Answer. If we assume that all projects are riskless and there are no transaction costs, then there should be one interest rate. Otherwise, it would be possible to make profits by borrowing money from banks with low interest rate and investing them in projects with high one (Zimmerman, 1997, p. 93). In such an ideal world this interest rate represents “the opportunity cost of capital” and if a project has positive net present value of its cash flows discounted at this rate, then its return is higher than ones generated by other investments (Zimmerman, 1997, p. 119). Therefore, it should be accepted. In reality calculating the net present value of a project one should use expected values of its uncertain cash flows “adjusted for inflation" and the interest rate calculated according to the following formula,

where is the average interest rate of the respective industry and is the inflation rate (Zimmerman, 1997, p. 108). Since the corporate income tax rate amounts to 34 % (Zimmerman, 1997, p. 109), it has significant influence on the cash flows of the project being evaluated. Therefore, the NPV method of evaluating the desirability of a project can be described as follows. A company should accept project with positive net present value of its expected “after-tax cash flows” adjusted for inflation and discounted using the interest rate calculated according to equation (1) (Zimmerman, 1997, p. 119). Two strengths of this method can be identified. First, it takes into account the time value of money, inflation and tax effects. Second, the interest rate at which cash flows are discounted in this method treats properly their risk. As for weaknesses, there are two of them, too. Firstly, this method does not recognize the moment when net present value of the stream of the project cash flows starts being positive (Money-Zine.com, n. d.). Since future cash flows are very uncertain (Zimmerman, 1997, p. 112), it is rather important. Secondly, since this method ignores the size of the investment, it is not very useful when “resources are scarce” (Money-Zine.com, n. d.).

2. Explain the PB period statistic, identify the PB’s acceptance benchmark, and identify what problem of the PB period method is corrected by using the discounted PB period method.

Answer. Payback statistics is the length of the time period “it takes to return the initial investment”. Contrary to the NPV method, there is no inherent acceptance benchmark in the payback “method of evaluating projects” (Zimmerman, 1997, p. 112). Therefore, it is up to the companies to establish such benchmarks for their projects. To explain the PB period statistic, let us consider the following stream of cash flows that consists of one outflow and four inflows. The first flow is the initial investment that equals –$300,000. In the subsequent four years each of the inflows equals $ 120,000. In this method it is assumed that each of the project’s cash inflows is spread evenly over the year at which it takes place. After two years $ 240,000 is going to be repaid. Additional $ 60,000 is going to be received during subsequent six months. Indeed, the yearly cash inflow in the amount of $120,000 can be decomposed in twelve monthly ones in the amount of $10,000 each. Therefore, the payback period statistics, in this case, is two and a half years. If the cost of capital in the industry in which the project is undertaken is 10 % per year, then the amounts of yearly cash inflows during the four year period are going to be the following: , , , . After three years, is going to be repaid. Additional is going to be received during subsequent days. Therefore, the discounted payback period statistics, in this case, is three years and a week (Money-Zine.com, n. d.). It is believed that cash flows scheduled beyond four years interval are very uncertain (Zimmerman, 1997, p. 112). Therefore, four years is the reasonable value for the PB’s acceptance benchmark. One of the drawbacks of the simple payback method is that it does not take into account the time value of money and the risk of project cash flows. At the same time, the discounted payback method does not have it (Money-Zine.com, n. d.).

3. Describe the Internal Rate of Return (IRR) method for determining a capital budgeting project’s desirability, identify the IRR's acceptance benchmark, and explain how NPV and IRR methods are similar and how they are different.

Answer. A stream of project's cash flows usually consists of an initial investment and subsequent future cash inflows. Using IRR method one is supposed to find the interest rate that turns the net value of these cash flows at the moment of the investment payment to zero. For such the ordinary streams, if the found rate is higher than some threshold rate, then the project should be undertaken, otherwise it should be rejected (Zimmerman 1997, p. 114). However, if the project's stream of cash flows has not ordinary structure, then one is supposed to find the set of values of the interest rate at which the net value of these cash flows at the moment of investment payment is greater than zero. If the chosen benchmark rae belongs to this set, then the project should be undertaken, otherwise it should be rejected (Zimmerman, 1997, pp. 116–117). Similarly to the NPV method, the average interest rate in the industry in which the project is undertaken can be chosen as such the threshold rate. Another option is to choose the “project’s cost of capital”. The last rule of accepting the project with unusual structure of its stream of cash outflows is valid for ordinary streams of cash flows as well. Indeed, since for any future cash inflow the discounting factor is strictly monotonically decreasing function of the interest rate (Zimmerman, 1997, pp. 95–102), the Net Discounted Value (NDV) of all cash flows of the ordinary stream has the following form: , where is an absolute value of the investment, and is a strictly monotonically decreasing function of interest rate . This form assumes the only value of the interest rate at which NDV is zero. In what follows it is denoted as . Besides, from this form it follows that for all interest rates greater than NDV is going to be negative, and for all interest rates smaller than NDV is going to be positive. Therefore, if the chosen benchmark interest rate is smaller than , then it belongs to the set of interest rate values at which NDV is positive, otherwise it does not belong to this set. Thus, the above mentioned rule is valid for any stream of cash flows. It should be noted that if the interest rate is equal to the industry average cost of capital, then the net discounted value of the project’s cash flows is equal to the project’s NPV. Therefore, if the project has positive NPV, then the industry average cost of capital belongs to the set of interest rate values at which NDV is positive. Moreover, if the industry average cost of capital belongs to the set of interest rate values at which its NDV is positive, then the project’s NPV is positive. Therefore, assuming that the industry average interest rate is chosen as a threshold rate in the IRR method, the one that is guided by IRR benchmark accepting projects and the one that uses the NPV benchmark are going to choose the same projects. However, a higher value of the project’s IRR does not imply a higher value of its NPV, and a higher value of the project’s NPV does not imply a higher value of its IRR (Zimmerman, 1997, p. 115).

4. Describe the Modified Internal Rate of Return (MIRR) method for determining a capital budgeting project’s desirability, identify the MIRR’s strengths and weaknesses, and explain the difference in the reinvestment rate assumption that distinguishes the MIRR from the IRR.

Answer. To calculate MIRR, firstly, one needs to find the value of the project’s outlays PV at the moment of the first project’s cash flow taking place using financing cost as an interest rate. Secondly, he\she is supposed to calculate the value of the project’s inflows FV at the moment of the project’s last cash flow taking place using firm’s cost of capital as an interest rate. The MIRR is calculated according to the following formula: where is the number of compounding periods in the length of the project. In this method the project is accepted if the chosen threshold interest rate is smaller than MIRR, otherwise it is rejected (“Modified Internal”, n. d.). As for the strengths of this method, it as well as the IRR one takes into account the time value of money (Money-Zine.com, n. d.). If the industry average interest rate is chosen as the benchmark rate, then this method recognizes the risk of the project's cash flows. This method supplements the NPV one in the following way. One that uses the NPV method can accept projects with relatively small positive NPV that assumes huge amount of investments. Such projects are likely to have small value of MIRR and, therefore, can be rejected by the method at hand. Thus, the MIRR method as well as IRR one is very useful when the resources are scarce. As for its weaknesses, the interest rates that can be chosen as the threshold rate tend to be volatile. It hinders the decision making a lot. To find the NPV of a bunch of projects, you need to add their NPVs. However, if you want to apply the MIRR method for evaluation of such the project, you need to repeat all calculations from the very beginning (Money-Zine.com, n. d.). Besides, using the method at hand, one can accept projects that assume small values of investments and NPVs (Zimmerman, 1997, p. 115). One of the IRR method drawbacks is the “reinvestment rate” assumption. Specifically, in this method high values of IRR assume that investment opportunities that provide same interest rate and have the same level of risk are readily available on the market (Zimmerman, 1997, p. 117). Since in the MIRR method the reinvestment rate is equal to the firm’s cost of capital (“Modified Internal”, n. d.), this drawback is absent in it.

5. Compute the NPV statistic for Project Y and tell [advise] whether the firm should accept or reject the project with the cash flows shown below if the appropriate cost of capital is 12 percent. Calculate the NPV statistic for a project, showing applicable input values, computational steps and formulas, and determine whether the firm should accept or reject the project. Explain how decreases in the cost of capital lead to an increase in the number of approved projects.

Project Y

Time

0

1

2

3

4

Cash Flow

-$11,000

$3,350

$4,180

$1,520

$2,000

Answer. The NPV of the stream of cash flows presented in the table is calculated according to the following formula: , where , , , , , (Money-Zine.com, n. d.). To calculate NPV, one needs to calculate the following values: , , , , Finally, . Since NPV is negative, the project should be rejected (Zimmerman, 1997).

Stream of the project's cash flows usually consists of initial investment and subsequent future cash inflows. Since for any future cash inflow the discounting factor is strictly monotonically decreasing function of the interest rate (Zimmerman, 1997, pp. 95–102), the NDV of all cash flows of the ordinary stream has the following form: , where is the absolute value of the investment, and is a strictly monotonically decreasing function of interest rate . If equals to the cost of capital in the industry in which the project being analyzed is undertaken, then NDV turns into NPV of this project (Money-Zine.com, n. d.). Therefore, NPV of the project with an ordinary stream of its cash flows is a strictly monotonically decreasing function of the cost of capital in the respective industry. Thus, decreases in the cost of capital entail increases in NPVs of the projects with ordinary streams of cash flows. Since most projects have ordinary streams of cash flows, a decrease in the cost of capital leads to an increase in the number of accepted projects.

6. Compute the payback period statistic for Project B and decide whether the firm should accept or reject the project with the cash flows shown below if the maximum allowable payback is three years. Calculate the PB period statistic, showing applicable input values, computational steps and formulas, and determine whether the project should be accepted or rejected, based on the firm’s maximum allowable PB period. Explain why, if the discounted payback period were computed, it would be less than, equal to, or greater than the non-discounted payback period.

Project B

Time

0

1

2

3

4

5

Cash Flow

-$11,000

$3,350

$4,180

$1,520

$950

$1,000

Answer. In project B initial investment equals $ 11,000. After two years, the following dollar amount is going to be repaid . After three years, the repaid dollar amount is going to be the following: . After four years, the repaid dollar amount is going to be the following: . Finally, after five years, the repaid dollar amount is going to be the following: . Since is equal to the initial investment, the PB period statistic is five years (Money-Zine.com, n. d.). For any future cash inflow the discounting factor is less than one (Zimmerman, 1997, pp. 95–102). A stream of project's cash flows usually consists of initial investment and subsequent future cash inflows. Therefore, for the project with an ordinary stream of cash flows the discounted payback period is always going to be greater than the non-discounted one. However, this can be not the case for projects with unusual stream of cash outflows. For instance, let us consider the project that starts with the cash inflow in the amount of $1,800. During the subsequent year, this project assumes the cash outlay in the amount of $2,000. In the following four year period the yearly cash inflows are scheduled in the amount of $200. After three years, the repaid amount is equal to $2,000. Since the only cash outlay in this project takes place in the second year and is equal to $2,000, the payback period is equal to 3 years. However, if the cost of capital equals 11.11 %, then the value of this cash outlay at the moment of the first cash flow is given by the following expression: . Therefore, the discounted payback period in this case is equal to 2 years. Thus, for an ordinary cash flow stream the discounted payback period is always greater than the non-discounted one. However, if structure of a cash flow stream assumes cash inflows followed by cash outflows, then this can be not the case.